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Chapter 8: Problem 19

Use de Moivre's Theorem to find each of the following. Write your answer in standard form. $$ \left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{6} $$

Short Answer

Expert verified
The expression is \( 32 + 32\sqrt{3}i \).

Step by step solution

01

Express in Polar Form

Start by recognizing the given expression in polar form: \[ z = 2(\cos 10^{\circ} + i\sin 10^{\circ}) \]This can be written as \( z = 2\text{cis}10^{\circ} \), where "cis" denotes \( \cos + i\sin \).
02

Apply de Moivre's Theorem

According to de Moivre鈥檚 Theorem, for a complex number in polar form \( r(\cos \theta + i \sin \theta) \), its nth power is:\[ z^n = r^n \left(\cos(n\theta) + i\sin(n\theta)\right) \]Apply it to \( z^6 \), where \( z = 2\text{cis}10^{\circ} \).
03

Calculate Magnitude

The magnitude \( r^6 \) is:\[ 2^6 = 64 \]Thus, the expression becomes:\[ 64\text{cis}(60^{\circ}) \]
04

Convert to Standard Form

Convert back to standard form:\[ 64\left( \cos 60^{\circ} + i \sin 60^{\circ} \right) \]Substitute the trigonometric values:\[ 64\left( \frac{1}{2} + i\frac{\sqrt{3}}{2} \right) \]
05

Simplify

Multiply through by 64:\[ 32 + 32\sqrt{3}i \]This is the expression in standard form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers form a fundamental concept in mathematics and can be quite intriguing. They combine a real part and an imaginary part into a single number, typically expressed as:
  • For example, a complex number is written as \( z = a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
  • The imaginary unit \( i \) satisfies the equation \( i^2 = -1 \).
Complex numbers are essential for various mathematical fields, providing insights and solutions to problems that real numbers alone cannot handle. They are especially useful in electrical engineering, quantum physics, and applied mathematics.
Polar Form
Polar form provides a unique and beneficial way to express complex numbers. In this form, a complex number is represented based on its distance from the origin (magnitude) and the angle it makes with the positive x-axis (argument).
  • The polar form of a complex number \( z = r(\cos \theta + i \sin \theta) \) can also be written as \( z = r\text{cis}\theta \), where "cis" stands for "cosine plus i sine."
  • The magnitude \( r \) is defined as \( \sqrt{a^2 + b^2} \) and represents the modulus of the complex number.
  • The argument \( \theta \) represents the angle, which can be found using trigonometric functions such as the arctan function.
Using the polar form can simplify the multiplication, division, and finding powers of complex numbers, especially through frameworks like de Moivre's Theorem.
Standard Form Conversion
Converting a complex number back into its standard form involves calculating its real and imaginary components from the polar form. The process typically involves using trigonometric identities to find each part:
  • Start with a complex number in polar form: \( z = r(\cos \theta + i \sin \theta) \).
  • Convert it by distributing the magnitude \( r \) to both the cosine and sine components.
  • For example, \( r(\cos \theta) = a \) becomes the real part, and \( r(i \sin \theta) = bi \) becomes the imaginary part.
This conversion makes it easier to interpret complex numbers and perform operations like addition and subtraction, which are more straightforward in standard form.
Trigonometric Functions
Trigonometric functions like sine and cosine are key components in both the polar form and the standard form conversion of complex numbers.
  • The cosine function, \( \cos \theta \), gives the horizontal (or x-axis) component of the distance from the origin.
  • The sine function, \( \sin \theta \), provides the vertical (or y-axis) component.
  • Trigonometric functions are periodic, repeating their values in cycles, which is crucial for simplifying expressions involving repeated angles.
These functions allow us to connect angles and distances in complex numbers, making them an integral part of expressing and manipulating these numbers in different forms.

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Most popular questions from this chapter

Use the formula for \(\cos (A+B)\) to find the exact value of \(\cos 75^{\circ}\).

Let \(\sin A=3 / 5\) with \(A\) in QI and \(\sin B=5 / 13\) with \(B\) in QI and find\(\sin (A+B)\)

Use the formula for \(\sin (A+B)\) to find the exact value of \(\sin 75^{\circ}\).

Write each complex number in trigonometric form, once using degrees and once using radians. In each case, begin by sketching the graph to help find the argument \(\theta\).\(-8 i\)

Write each complex number in trigonometric form, once using degrees and once using radians. In each case, begin by sketching the graph to help find the argument \(\theta\).\(-1+i\)
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